Let
B_{r}(E)
B
r
(
E
)
be the closed ball of radius
r
r
around the origin in a real Banach space
E
E
and
\mathcal{F}_{r}(E)
F
r
(
E
)
be the set of all
r
r
-Lipschitz continuous convex functions defined on
B_{r}(E)
B
r
(
E
)
. Suppose
f
f
is a real-valued and bounded below function on
B_{r}(E)
B
r
(
E
)
. We define the
I
I
-conjugate function
f^{I}
f
I
of
f
f
to improve the Fenchel inequality and investigate the properties of
f^{I}
f
I
. In particular,
(f^{I})^{I}
(
f
I
)
I
coincides with
f
f
on
B_{r}(E)
B
r
(
E
)
if and only if
f
f
is in
\mathcal{F}_{r}(E)
F
r
(
E
)
. Excluding the value
+\infty
+
∞
, the transformation from
f
f
to
f^{I}
f
I
enlarges the potentiality of the contribution to numerical computation for convex analysis.